A205488 -id:A205488

A205476

G.f.: exp( Sum_{n>=1} (x^n/n) * Product_{d|n} (1 + n*x^d/d) ).

+10
8

1, 1, 2, 3, 5, 8, 12, 20, 28, 45, 65, 101, 148, 221, 316, 469, 673, 969, 1420, 2025, 2892, 4100, 5905, 8314, 11860, 16645, 23399, 32838, 46071, 64274, 89761, 124977, 173231, 240492, 332978, 460015, 634271, 874464, 1200463, 1649499, 2263102, 3098661, 4239109

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,3

COMMENTS

Note: exp( Sum_{n>=1} (x^n/n) * Product_{d|n} (1 + x^d) ) does not yield an integer series.

LINKS

Table of n, a(n) for n=0..42.

FORMULA

Logarithmic derivative yields A205477.

EXAMPLE

G.f.: A(x) = 1 + x + 2*x^2 + 3*x^3 + 5*x^4 + 8*x^5 + 12*x^6 + 20*x^7 + ...

By definition:

log(A(x)) = x*(1+x) + x^2*(1+2*x)*(1+x^2)/2 + x^3*(1+3*x)*(1+x^3)/3 + x^4*(1+4*x)*(1+2*x^2)*(1+x^4)/4 + x^5*(1+5*x)*(1+x^5)/5 + x^6*(1+6*x)*(1+3*x^2)*(1+2*x^3)*(1+x^6)/6 + ...

Explicitly,

log(A(x)) = x + 3*x^2/2 + 4*x^3/3 + 7*x^4/4 + 11*x^5/5 + 12*x^6/6 + 29*x^7/7 + 15*x^8/8 + 49*x^9/9 + 43*x^10/10 + ... + A205477(n)*x^n/n + ...

MATHEMATICA

max = 50; s = Exp[Sum[(x^n/n)*Product[1+n*x^d/d, {d, Divisors[n]}], {n, 1, max}]] + O[x]^max; CoefficientList[s , x] (* Jean-François Alcover, Dec 23 2015 *)

PROG

(PARI) {a(n)=polcoeff(exp(sum(m=1, n+1, x^m/m*exp(sumdiv(m, d, log(1+m*x^d/d+x*O(x^n)))))), n)}

CROSSREFS

Cf. A205477 (log), A198304, A205478, A205480, A205482, A205484, A205486, A205488, A205490.

KEYWORD

nonn

AUTHOR

Paul D. Hanna, Jan 27 2012

STATUS

approved

A205478

G.f.: exp( Sum_{n>=1} (x^n/n) * Product_{d|n} (1 + d*x^n) ).

+10
8

1, 1, 2, 2, 4, 4, 8, 8, 14, 15, 24, 25, 43, 45, 69, 74, 113, 120, 187, 198, 291, 314, 452, 483, 720, 770, 1089, 1182, 1657, 1784, 2530, 2724, 3764, 4102, 5593, 6053, 8361, 9049, 12183, 13304, 17831, 19378, 26097, 28355, 37548, 41107, 54031, 58894, 78008, 85052

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,3

COMMENTS

Note: exp( Sum_{n>=1} (x^n/n) * Product_{d|n} (1 + x^n) ) does not yield an integer series.

LINKS

Table of n, a(n) for n=0..49.

FORMULA

Logarithmic derivative yields A205479.

EXAMPLE

G.f.: A(x) = 1 + x + 2*x^2 + 2*x^3 + 4*x^4 + 4*x^5 + 8*x^6 + 8*x^7 + ...

By definition:

log(A(x)) = x*(1+x) + x^2*(1+x^2)*(1+2*x^2)/2 + x^3*(1+x^3)*(1+3*x^3)/3 + x^4*(1+x^4)*(1+2*x^4)*(1+4*x^4)/4 + x^5*(1+x^5)*(1+5*x^5)/5 + x^6*(1+x^6)*(1+2*x^6)*(1+3*x^6)*(1+6*x^6)/6 + ...

Explicitly,

log(A(x)) = x + 3*x^2/2 + x^3/3 + 7*x^4/4 + x^5/5 + 15*x^6/6 + x^7/7 + 15*x^8/8 + 10*x^9/9 + 13*x^10/10 + ... + A205479(n)*x^n/n + ...

MATHEMATICA

max = 50; s = Exp[Sum[(x^n/n)*Product[1+d*x^n, {d, Divisors[n]}], {n, 1, max}]] + O[x]^max; CoefficientList[s, x] (* Jean-François Alcover, Dec 23 2015 *)

PROG

(PARI) {a(n)=polcoeff(exp(sum(m=1, n+1, x^m/m*exp(sumdiv(m, d, log(1+d*x^m+x*O(x^n)))))), n)}

CROSSREFS

Cf. A205479, A198296, A205476, A205480, A205482, A205484, A205486, A205488, A205490.

KEYWORD

nonn

AUTHOR

Paul D. Hanna, Jan 27 2012

STATUS

approved

A205480

G.f: exp( Sum_{n>=1} x^n/n * Product_{d|n} (1 + d*x^(n/d))^d ).

+10
8

1, 1, 2, 4, 10, 27, 76, 242, 852, 3016, 11262, 47004, 204761, 894673, 4134909, 20370101, 101904474, 521459464, 2813783214, 15616060213, 87143803196, 502477538546, 3039137586808, 18763942581733, 116737580008529, 742909490860950, 4846956807516551

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,3

COMMENTS

Note: exp( Sum_{n>=1} x^n/n * Product_{d|n} (1 + x^(n/d))^d ) does not yield an integer series.

LINKS

Table of n, a(n) for n=0..26.

FORMULA

Logarithmic derivative yields A205481.

EXAMPLE

G.f.: A(x) = 1 + x + 2*x^2 + 4*x^3 + 10*x^4 + 27*x^5 + 76*x^6 + 242*x^7 + ...

By definition:

log(A(x)) = x*(1+x) + x^2*(1+x^2)*(1+2*x)^2/2 + x^3*(1+x^3)*(1+3*x)^3/3 + x^4*(1+x^4)*(1+2*x^2)^2*(1+4*x)^4/4 + x^5*(1+x^5)*(1+5*x)^5/5 + x^6*(1+x^6)*(1+2*x^3)^2*(1+3*x^2)^3*(1+6*x)^6/6 + ...

Explicitly,

log(A(x)) = x + 3*x^2/2 + 7*x^3/3 + 23*x^4/4 + 76*x^5/5 + 249*x^6/6 + 974*x^7/7 + 4151*x^8/8 + 16558*x^9/9 + ... + A205481(n)*x^n/n + ...

MATHEMATICA

max = 30; s = Exp[Sum[(x^n/n)*Product[(1+d*x^(n/d))^d, {d, Divisors[n]}], {n, 1, max}]] + O[x]^max; CoefficientList[s , x] (* Jean-François Alcover, Dec 23 2015 *)

PROG

(PARI) {a(n)=polcoeff(exp(sum(m=1, n+1, x^m/m*exp(sumdiv(m, d, d*log(1+d*x^(m/d)+x*O(x^n)))))), n)}

CROSSREFS

Cf. A205481 (log), A205476, A205478, A205482, A205484, A205486, A205488, A205490.

KEYWORD

nonn

AUTHOR

Paul D. Hanna, Jan 27 2012

STATUS

approved

A205482

G.f.: exp( Sum_{n>=1} x^n/n * Product_{d|n} (1 + d*x^n)^d ).

+10
8

1, 1, 2, 2, 5, 5, 15, 15, 34, 46, 84, 96, 246, 282, 512, 696, 1421, 1713, 3436, 4084, 8227, 10821, 19128, 23258, 48474, 60943, 106780, 139313, 252748, 322577, 600660, 760872, 1365570, 1807979, 3064882, 3951491, 7358684, 9476993, 15962935, 21243381

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,3

COMMENTS

Note: exp( Sum_{n>=1} x^n/n * Product_{d|n} (1 + x^n)^d ) does not yield an integer series.

LINKS

Table of n, a(n) for n=0..39.

FORMULA

Logarithmic derivative yields A205483.

EXAMPLE

G.f.: A(x) = 1 + x + 2*x^2 + 2*x^3 + 5*x^4 + 5*x^5 + 15*x^6 + 15*x^7 + ...

By definition:

log(A(x)) = x*(1+x) + x^2*(1+x^2)*(1+2*x^2)^2/2 + x^3*(1+x^3)*(1+3*x^3)^3/3 + x^4*(1+x^4)*(1+2*x^4)^2*(1+4*x^4)^4/4 + x^5*(1+x^5)*(1+5*x^5)^5/5 + x^6*(1+x^6)*(1+2*x^6)^2*(1+3*x^6)^3*(1+6*x^6)^6/6 + ...

Explicitly,

log(A(x)) = x + 3*x^2/2 + x^3/3 + 11*x^4/4 + x^5/5 + 45*x^6/6 + x^7/7 + 59*x^8/8 + 109*x^9/9 + 53*x^10/10 + ... + A205483(n)*x^n/n + ...

MATHEMATICA

max = 40; s = Exp[Sum[(x^n/n)*Product[(1 + d*x^n)^d, {d, Divisors[n]}], {n, 1, max}]] + O[x]^max; CoefficientList[s , x] (* Jean-François Alcover, Dec 23 2015 *)

PROG

(PARI) {a(n)=polcoeff(exp(sum(m=1, n+1, x^m/m*exp(sumdiv(m, d, d*log(1+d*x^m+x*O(x^n)))))), n)}

CROSSREFS

Cf. A205483 (log), A205476, A205478, A205480, A205484, A205486, A205488, A205490.

KEYWORD

nonn

AUTHOR

Paul D. Hanna, Jan 27 2012

STATUS

approved

A205484

G.f.: exp( Sum_{n>=1} x^n/n * Product_{d|n} (1 + d*x^d)^n ).

+10
8

1, 1, 2, 3, 7, 14, 30, 65, 132, 280, 632, 1439, 3299, 7569, 17450, 40313, 92889, 212801, 483590, 1092649, 2467078, 5581232, 12690828, 29123728, 67648617, 159370347, 381080620, 923803158, 2264970530, 5599185887, 13909201590, 34612152762, 86049014990

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,3

COMMENTS

Note: exp( Sum_{n>=1} x^n/n * Product_{d|n} (1 + x^d)^n ) does not yield an integer series.

LINKS

Table of n, a(n) for n=0..32.

FORMULA

Logarithmic derivative yields A205485.

EXAMPLE

G.f.: A(x) = 1 + x + 2*x^2 + 3*x^3 + 7*x^4 + 14*x^5 + 30*x^6 + 65*x^7 + ...

By definition:

log(A(x)) = x*(1+x) + x^2*(1+x)^2*(1+2*x^2)^2/2 + x^3*(1+x)^3*(1+3*x^3)^3/3 + x^4*(1+x)^4*(1+2*x^2)^4*(1+4*x^4)^4/4 + x^5*(1+x)^5*(1+5*x^5)^5/5 + x^6*(1+x)^6*(1+2*x^2)^6*(1+3*x^3)^6*(1+6*x^6)^6/6 + ...

Explicitly,

log(A(x)) = x + 3*x^2/2 + 4*x^3/3 + 15*x^4/4 + 31*x^5/5 + 72*x^6/6 + 176*x^7/7 + 327*x^8/8 + 751*x^9/9 + ... + A205485(n)*x^n/n + ...

MATHEMATICA

max = 40; s = Exp[Sum[(x^n/n)*Product[(1 + d*x^d)^n, {d, Divisors[n]}], {n, 1, max}]] + O[x]^max; CoefficientList[s , x] (* Jean-François Alcover, Dec 23 2015 *)

PROG

(PARI) {a(n)=polcoeff(exp(sum(m=1, n+1, x^m/m*exp(sumdiv(m, d, m*log(1+d*x^d+x*O(x^n)))))), n)}

CROSSREFS

Cf. A205485 (log), A205476, A205478, A205480, A205482, A205486, A205488, A205490.

KEYWORD

nonn

AUTHOR

Paul D. Hanna, Jan 27 2012

STATUS

approved

A205486

G.f.: exp( Sum_{n>=1} (x^n/n) / Product_{d|n} (1 - d*x^(n/d))^d ).

+10
8

1, 1, 2, 5, 16, 60, 259, 1273, 7048, 43241, 289685, 2097912, 16317134, 135574160, 1196898329, 11168544771, 109647222799, 1128440311914, 12139734936953, 136195813530558, 1590028534430967, 19277087785530470, 242235954813757132, 3149491477171141810

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,3

COMMENTS

Note: exp( Sum_{n>=1} (x^n/n) / Product_{d|n} (1 - x^(n/d))^d ) does not yield an integer series.

LINKS

Table of n, a(n) for n=0..23.

FORMULA

Logarithmic derivative yields A205487.

EXAMPLE

G.f.: A(x) = 1 + x + 2*x^2 + 5*x^3 + 16*x^4 + 60*x^5 + 259*x^6 + 1273*x^7 +...

By definition:

log(A(x)) = x/(1-x) + (x^2/2)/((1-x^2)*(1-2*x)^2) + (x^3/3)/((1-x^3)*(1-3*x)^3) + (x^4/4)/((1-x^4)*(1-2*x^2)^2*(1-4*x)^4) + (x^5/5)/((1-x^5)*(1-5*x)^5) + (x^6/6)/((1-x^6)*(1-2*x^3)^2*(1-3*x^2)^3*(1-6*x)^6) +...

Explicitly,

log(A(x)) = x + 3*x^2/2 + 10*x^3/3 + 43*x^4/4 + 206*x^5/5 + 1104*x^6/6 + 6581*x^7/7 + 43227*x^8/8 +...+ A205487(n)*x^n/n +...

PROG

(PARI) {a(n)=polcoeff(exp(sum(m=1, n+1, x^m/m*exp(sumdiv(m, d, -d*log(1-d*x^(m/d)+x*O(x^n)))))), n)}

CROSSREFS

Cf. A205487 (log), A205476, A205478, A205480, A205482, A205484, A205488, A205490.

KEYWORD

nonn

AUTHOR

Paul D. Hanna, Jan 27 2012

STATUS

approved

A205489

L.g.f.: Sum_{n>=1} (x^n/n) / Product_{d|n} (1 - d*x^n)^d.

+10
8

1, 3, 4, 15, 6, 78, 8, 247, 202, 708, 12, 4146, 14, 5498, 8964, 24135, 18, 81114, 20, 206520, 193736, 225558, 24, 2314378, 242656, 1278332, 3622954, 9209950, 30, 26654118, 32, 58890983, 59213598, 35652216, 28736938, 628796418, 38, 179307278, 878319368

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

LINKS

Table of n, a(n) for n=1..39.

FORMULA

Forms the logarithmic derivative of A205488.

EXAMPLE

L.g.f.: L(x) = x + 3*x^2/2 + 4*x^3/3 + 15*x^4/4 + 6*x^5/5 + 78*x^6/6 +...

By definition:

L(x) = x/(1-x) + (x^2/2)/((1-x^2)*(1-2*x^2)^2) + (x^3/3)/((1-x^3)*(1-3*x^3)^3) + (x^4/4)/((1-x^4)*(1-2*x^4)^2*(1-4*x^4)^4) + (x^5/5)/((1-x^5)*(1-5*x^5)^5) + (x^6/6)/((1-x^6)*(1-2*x^6)^2*(1-3*x^6)^3*(1-6*x^6)^6) +...

Exponentiation yields the g.f. of A205488:

exp(L(x)) = 1 + x + 2*x^2 + 3*x^3 + 7*x^4 + 9*x^5 + 26*x^6 + 32*x^7 +...

PROG

(PARI) {a(n)=n*polcoeff(sum(m=1, n+1, x^m/m*exp(sumdiv(m, d, -d*log(1-d*x^m+x*O(x^n))))), n)}

CROSSREFS

Cf. A205488 (exp), A205477, A205479, A205481, A205483, A205485, A205487, A205491.

KEYWORD

nonn

AUTHOR

Paul D. Hanna, Jan 27 2012

STATUS

approved

A205490

G.f.: exp( Sum_{n>=1} (x^n/n) / Product_{d|n} (1 - d*x^d)^n ).

+10
8

1, 1, 2, 4, 10, 22, 57, 134, 331, 797, 1995, 4879, 12367, 31056, 79315, 202370, 521575, 1339934, 3456778, 8885907, 22848211, 58576714, 150117209, 384135566, 983789032, 2522109065, 6485104365, 16736092434, 43408268497, 113201300205, 296975753940, 783578962587

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,3

COMMENTS

Note: exp( Sum_{n>=1} (x^n/n) / Product_{d|n} (1 - x^d)^n ) does not yield an integer series.

LINKS

Table of n, a(n) for n=0..31.

FORMULA

Logarithmic derivative yields A205491.

EXAMPLE

G.f.: A(x) = 1 + x + 2*x^2 + 4*x^3 + 10*x^4 + 22*x^5 + 57*x^6 + 134*x^7 +...

By definition:

log(A(x)) = x/(1-x) + (x^2/2)/((1-x)^2*(1-2*x^2)^2) + (x^3/3)/((1-x)^3*(1-3*x^3)^3) + (x^4/4)/((1-x)^4*(1-2*x^2)^4*(1-4*x^4)^4) + (x^5/5)/((1-x)^5*(1-5*x^5)^5) + (x^6/6)/((1-x)^6*(1-2*x^2)^6*(1-3*x^3)^6*(1-6*x^6)^6) +...

Explicitly,

log(A(x)) = x + 3*x^2/2 + 7*x^3/3 + 23*x^4/4 + 51*x^5/5 + 165*x^6/6 + 386*x^7/7 + 1039*x^8/8 + 2554*x^9/9 +...+ A205491(n)*x^n/n +...

PROG

(PARI) {a(n)=polcoeff(exp(sum(m=1, n+1, x^m/m*exp(sumdiv(m, d, -m*log(1-d*x^d+x*O(x^n)))))), n)}

CROSSREFS

Cf. A205491 (log), A205476, A205478, A205480, A205482, A205484, A205486, A205488.

KEYWORD

nonn

AUTHOR

Paul D. Hanna, Jan 27 2012

STATUS

approved